证明:若无穷积分绝对收敛,函数φ（x)在[a,+∞)单调有界,则无穷积分收敛.

证明:若函数f(x)在a连续,则函数 <img src='https://img2.soutiyun.com/ask/2020-11-11/973957882099598.png' /> 在a都连续.

证明:若在[a,∞;c,d]内成立|f(x,y)|≤F(x,y)， 并且关于y∈[c,d]积分<img src='https://img2.soutiyun.com/ask/2021-01-27/980588087069898.png' />关于y∈[c,d], 亦一致收敛，且绝对收敛

证明:若函数f(x)在区间[a,+∞)上连续且有极限<img src='https://img2.soutiyun.com/ask/2020-12-13/976732708656138.png' />则(x)在区间[a,+∞)上是有界的.

定义在[a，b]上的无界函数f(x)的收敛，积分<img src='https://img2.soutiyun.com/latex/latex.action' />是否可以视为相应积分和数<img src='https://img2.soutiyun.com/latex/latex.action' />(这里x_i≤ξ_i≤x_i+1且△x_i=x_i+1-x_i)的极限？

证明:若函数f(x)＞0,在[a,b]可积,令<img src='https://img2.soutiyun.com/ask/2020-11-13/974108578848118.png' />则 <img src='https://img2.soutiyun.com/ask/2020-11-13/974108619490443.png' />

证明:若函数f(x)与φ(x)在[a,b]连续,则 <img src='https://img2.soutiyun.com/ask/2020-11-12/974060778329609.png' />

证明:若函数f(x)在(a,+∞)连续,且<img src='https://img2.soutiyun.com/ask/2020-11-11/97395757022676.png' />则f(x)在(a,+∞)有界.

A.<img src='https://img2.soutiyun.com/ask/2019-07-12/931777046302068.png' /> B.<img src='https://img2.soutiyun.com/ask/2019-07-12/931777058174924.png' /> C.<img src='https://img2.soutiyun.com/ask/2019-07-12/931777073741383.png' /> D.<img src='https://img2.soutiyun.com/ask/2019-07-12/931777085020813.png' />

f(x，y)=φ(x)，(x，y)∈D=[a，b]×(-∞，+∞)． 试讨论f在D上是否连续？是否一致连续？

若函数f(x)在[a,b]上可积，证明存在折线函数列<img src='https://img2.soutiyun.com/ask/2021-01-22/98018065819823.png' />

证明:若函数f(x)在[a,b]单调增加,则 <img src='https://img2.soutiyun.com/ask/2020-11-12/974060207561962.png' />

证明:若级数<img src='https://img2.soutiyun.com/ask/2020-11-13/974117042967238.jpg' />绝对收敛,则函数项级数 <img src='https://img2.soutiyun.com/ask/2020-11-13/974117058462124.png' /> 在R一致收敛.

证明:0＜λ＜1,无穷积分<img src='https://img2.soutiyun.com/ask/2020-11-13/974141104841498.png' />都条件收敛.

设f(x,y)在[a,+∞;c,d]连续，对[c,d)上每一个<img src='https://img2.soutiyun.com/ask/2021-01-27/98058817504593.png' />收敛，但积分在y= d发散.证明这积分在[c,d]非一致收敛。

证明:(1)若函数f在[a,b]上可导,且f'(x)≥m,则 <img src='https://img2.soutiyun.com/ask/2021-02-04/98128598322409.png' /> (2)若函数f在[a,b]上可导,且 <img src='https://img2.soutiyun.com/ask/2021-02-04/981285989538451.png' /> (3)对任意实数x₁,x₂,都有 <img src='https://img2.soutiyun.com/ask/2021-02-04/981286001647143.png' />

若连续函数列{φ₀(x)，φ₁(x)，…}在[a，b]上带权ρ(x)正交，且<img src='https://img2.soutiyun.com/ask/2020-07-31/965040632752842.png' />内恒正，证明:,对任意n个数<img src='https://img2.soutiyun.com/ask/2020-07-31/965040650416852.png' />，广义多项式<img src='https://img2.soutiyun.com/ask/2020-07-31/965040667564833.png' />在[a，b]上至少有一个零点。